Low molecular weight polyethylene polymers fall under the category of polyethylene oligomers. The polyethylene oligomers are also referred to as a polyethylene wax, and are roughly classified into animal and plant-based, mineral-based, refined petroleum-based, and synthetic types. Examples of polyethylene waxes having a functional group at one terminal of a polymer chain thereof include a mineral-based montanic acid wax. However, the montanic acid wax has a problem of high production costs since it is extracted and purified from natural brown coal.
One example of a synthetic systems that enables production of a polyethylene wax at a low cost is a system using a metallocene catalyst. However, in such a system, although a, reactive vinyl group can be introduced into one terminal of the polymer chain, a chain transfer agent such as hydrogen needs to be used for reducing the molecular weight of the polymer chain. Since the terminal of a polymer generated by the chain transfer agent is saturated thereby, there has been a problem that the number of terminal vinyl groups of the polymer chain to which other functional groups can be introduced is extremely small.
Also, in either a high pressure radical method or a Ziegler catalyst method, both of which are widely-used synthetic processes, a terminal vinyl group of a polymer chain is difficult to obtain, and moreover, it is difficult to intentionally introduce the terminal vinyl group into the polymer chain.
Much research has been made into overcoming the above-described problems, and a method for introducing a double bond into a polymer chain with a high ratio has been developed (e.g., Japanese Patent Application Laid-Open (JP-A) No. 2003-73412). On the other hand, as a method for introducing a functional group to a double bond of a polymer chain, a method for modifying a liquid ethylene polymer having a double bond at one terminal thereof with an unsaturated carboxylic acid derivative or the like is known (e.g., Japanese Patent Publication (JP-B) No. 7-78098), and a method for modifying a double bond at one terminal of a low molecular weight ethylene polymer into a functional group of various kinds (e.g., JP-A No. 1-217007), and the like are also known.
However, these modification methods have problems in that the ratio of modification of a double bond into a functional group is low, purification of the resultant is laborious due to the simple polymerization of the unsaturated carboxylic acid derivative used in the method, or the like.    [Patent-Document 1] JP-A No. 2003-73412    [Patent Document 2] JP-B No. 7-78098    [Patent Document 3] JP-A No. 1-217007